Mathematical Abstraction as the Result of
a Delicate Shift of Attention
JOHN MASON
Students of mathematics often say that they find mathematics abstract, and give this as the reason for being
stuck, for disliking mathematics lessons, or even for withdrawing from mathematics altogether Yet the power of
mathematics, and the pleasme that mathematicians get
from it, arise from precisely the abstract nature of
mathematics The aim of this article is to explore this
unfortunate dichotomy of response to the idea of abstraction, to venture a technical use of the term which could be
of help to mathematics teachers and students alike, and to
provide a case study.
The first question is whether the word abstract is actually being used in the same way by frustrated students and
by inspired professionals The verb is usually pronounced
differently to the adjective and noun, emphasis being
placed on the prefix ah when it is used as a noun 01
adjective (as in the beginnings of papers, and when applied to an idea that seems unconnected with reality or
which fails to inspire confidence-hence the pejorative
meaning) and on the root stem 5/ract when used to refer to
a process akin to extracting Thus exttaa means to draw
out and to abstrau means to draw away Eco, in discussing the meaning of aesthetic in the work of St Thomas
Aquinas, writes that
Aesthetic seeing involves grasping the form in the
sensible. It therefore occurs prior to the act of
abstraction, because in abstraction the form is
divorced from the sensible. [Eco, 1988, page 193]
If emphasis is placed on divorced, we are reminded of the
student's experience It is easy to sympathise with the
student's sense of abstract as removed from or divorced
from reality (or perhaps, more accurately, from meaning,
since om reality consists in that which we find meaningful) But perhaps this sense of being out of contact arises
because there has been little or no participation in the
process of abstraction, in the movement of drawing away
Per haps all the students are a ware of is the having been
drawn away rather than the drawing itself
If emphasis is placed on form, we are reminded of the
expert's experience Tall [1988] speaks in a similar vein of
abstraction (as) the isolation of specific attributes of
a concept so that they can be considered separately
from the other attributes
When forms become objects or components of thought,
and when with familiarity they become mentally manipulable, becoming, as it were, concrete, mathematics finds
2
its greatest power But abstraction is not just higher mathematics for the few It is an integral part of speaking and
thinking. C S Peirce, the noted creator of pragmaticism,
having drawn attention to the abstractions which we call
collections, namely pairs, dozens, sonnets, scores, etc,
wrote:
the great rolling billows of abstraction in the
ocean of mathematical thought; but when we come
to minute examination of it, we shall find, in every
department, incessant ripples of the same form of
thought
[Peirce, 1982]
Far from being an abstruse activity of expert mathematicians, abstraction is a common experience Why then
does mathematical abstraction get the reaction it does?
My thesis is that the uses ofthe word abstract in mathematics by both novices and professionals refers to a common, root experience: an extremely brief moment which
happens in the twinkling of an eye; a delicate shift of
attention from seeing an expression as an expression of
generality, to seeing the expression as an object or property Thus, abstracting lies between the expression of
generality and the manipulation of that expression while,
for example, constructing a convincing argument In that
ever so delicate shift of attention occurs the drawing away
of form from the sensible, the abstraction, referred to in
the quotation from Aquinas
When the shift occurs, it is hardly noticeable and, to a
mathematician, it seems the most natm al and obvious
movement imaginable Consequently it fails to attract the
expert's attention When the shift does not occur, it
blocks progress and makes the student feel out of touch
and excluded, a mere observer in a peculiar ritual Some
students even master aspects of the form of the ritual
without being able to explain why they do what they do. A
few, through this process of habituation [Peirce, 1982;
Mason and Davis, 1988] find themselves able to explain
things to others, but many never completely lose their
sense of alienation
My approach to the thesis makes use of the theory of
shifts of attention [Mason and Davis, 1988] and proceeds
via the Discipline of Noticing [Mason, 1987] I present a
few episodes from a mathematical case study in which you
the reader are asked to participate. The case study is based
on part of what happened during a weekend of mathematical problem solving devoted to the theme of axioms and
abstracting, with participants varying from mathematically naive undergraduates to sophisticated graduates
For the Learning of Mathematics 9, 2 (June 1989)
Fl M Publishing Association. Montreal, Quebec . Canada
and tutors of the Open University.
By engaging in the mathematical tasks, it is my prediction that you will experience aspects of abstracting which
I have found significant By means of these examples, you
will (through the process of generalising from my examples) construe what I mean by various terms, such as shift
of attention, characterising, and abstrauing, and because
the descriptions are related to mathematical experience,
you will be more likely to notice their applicability in
mathematical moments in the future This in turn will
enable you, should you so wish, to take alternative action,
or at least to be more sharply aware of aspects which
might otherwise have been overlooked As St Augustine
wrote
In the halls of memory we bear the images of things
once perceived, as memorials which we can contemplate mentally, and can speak of with a good
conscience and without lying. But these memorials
belong to us privately. If anyone hears me speak of
them, provided he has seen them himself, he does
not learn from my words, but recognises the truth of
what I say by the images he has in his own memory
But if he has not had these sensations, obviously he
believes my words rather than learns from them
[St. Augustine, 389]
Even with the influence of a modern translation, this
observation ofSt Augustine neatly expresses the purpose
of my offering a case study, and the effect I intend it to
have The validity of my thesis lies in the extent to which
you recognise the aspects which I stress, and the extent to
which you find such awareness helpful in the future. This
methodology is itself a process of abstraction, moving as
it does from experience, to expressing experience, to taking such expressions as descriptions of properties of many
experiences, to manipulating labels of those experiences
in subsequent descriptions.
In Floyd eta/ [1982], a helix was used to describe the
experience of moving from manipulating objects (physical, pictorial, symbolic, mental) to getting-a-sense-of
some feature or property of those objects, to articulating
that property as an expression of generality, to finding
that expression becoming a confidence-inspiring entity
which can be manipulated and used to seek out further
properties. I suggest that the process of abstracting in
mathematics lies in the momentary movement from
articulating to manipulating
Getflng o sense of
r:=:=-~ 0 senseot
~
Articulation of a seeing of generality, first in words and
pictures, and then in increasingly tight and economically
succinct expressions, using symbols and perhaps diagrams, is a pinnacle of achievement, often achieved only
after great struggle. It turns into a mere foothill as it
becomes a staging post for further work with the expression as a manipulable object. The helix image not only
helps to locate abstracting in a flowing process, but also
reinforces the notion that what is abstract for one person
can become concrete and confidence-inspiring for others
[Mason, 1980]
The language of proces; and object is a little too glib,
for in the stressing which the words imply, there is a
tendency to ignore the experience of a sense-of, which
accompanies or is associated with the expression, and
which does not disappear when the expression becomes
an object of attention . On the contrary. The expression
acts as a signal to recall salient associations Just as there
is a huge difference between drawing your own figure in
order to stabilise your mental imagery and to extend your
thinking power, so there is a huge difference between
expressing your own generality and doing someone else's
algebra. Algebraic expressions provide a generic example
of how meaning can remain connected to symbols: it is
important, for algebraic thinking to develop effectively,
to maintain a dual awareness of expressions, as entities or
objects, and as statements about how a calculation is to be
performed [Mason, 1982] In the language of Tall and
Vinner [1981], the concept image is extended and made
more powerful by the manipulability of the expression,
not narrowed and refined
case study
A sequence Express to yourself in action (by doing
it) and in words (by talking to yourself or a colleague) a rule for continuing the following array
[Honsberger 1970 page 87]:
10
5
11
2
6
12
3
7
13
4
8
14
9
15
16
Now attend to the central line, and generate more
terms. Find a way of generating even more terms
without filling in the other numbers Express your
rule in general terms, and as a formula. Pause now
and try it.
Comment. The action of filling in more numbers, whether
performed physically or just mentally, serves to clarify
and crystallise a sense of what is going on, and prepares
the way for a verbal statement of a general rule For
example, the presence of the square numbers in certain
3
salient positions is usually noticed . Furthermme, the
instruction to attend to the central line already focuses
attention on certain aspects of the array, and away from
others. The instructions are intended to prompt a move
towards expressing a general rule, through the experience
of particular cases . Young students in their early exposure
to algebra can often express a general rule but find it
difficult to see that expression as a manipulable object.
The theory of shifts suggests that they need continued
exposure to such acts of expressing so that they begin to
find it relatively easy, and thereby find some of their
attention free to view formulae as objects Fmmulae are
abstract, in the pejmative sense, if they are drawn away
from 01 unconnected to meaning, or in other words, if
they fail to be confidence-inspiring as potentially manipulable objects
Observation Notice that if you start at the 3 and
count along a further .3 terms, a further 3, a further .3
still,
, those terms are all divisible by 3 Similarly if you start at 7 and count along from there 7,
14, 21,. . terms, the terms you reach are all divisible by 7 Will this always happen for this sequence?
Pause now and try it
Comment. The first task is to get a sense of the thi:~ in the
statement that might "always happen," and after. there is
more than one way to make sense of it The observer has
stressed certain features of the sequence, has edited out or
ignored others, and expects the reader to be stressing in
the same way But the reader construes this stressing by
responding to instructions, and then reconstructing the
generality which may or may not be what the observer has
already experienced
To answer the question convincingly, after having tried
numerous examples and becoming convinced that it does
wmk, it is useful to express the it that always happens in a
manipulable form, and then to manipulate it as part of an
algebraic argument. It is not at all easy in this case, even
for sophisticated students of mathematics, because it
involves a term as part of a subscript of a term. In other
wmds, an expression such as tn must have become confidently manipulable as an entity. The movement from a
formula for tn, to that formula seen as an entity, is precisely the delicate shift of attention which I associate with
abstracting The shift from looking-attn to working-with
tn is ever so slight, yet it constitutes a fundamental, and for
some people, difficult shift
During a different seminar in which the same sequence
was offered and participants invited to express the same
divisibility property, one person observed that:
Observation: If d divides tn, then d divides tn+d
There then followed a discussion as to whether this was
actually the same or stronger than version involving !n
dividing tn+ktn Fm some it was obvious that they said the
same thing, others saw them as completely different
expressions By specialising, we were able to convince
ourselves that the first implied the second but not vice
versa . It was then noted that in order to discuss whether
one statement implies another, there had to be another
4
abstraction shift: seeing the expressions (which until then
had been attempts to say what someone was seeing in the
given sequence) as objects of attention while looking at
logical connections between them. Peirce was right; there
are incessant ripples at every level
Extending. What is special about this particular
sequence I, 3, 7,
?
Comment. The question draws attention to an observation which might have been made during the proof of
either of the previous observations, that the form of
argument does not make use of the particular fmmula for
to, but only of the fact that it is a polynomial. Such an
observation is most likely to come from a sophisticated
mathematical thinker, because it requires a splitting of
attention, one part involved in calculation while another
part remains aloof but observant This is the monitor on
the shoulder [Mason, Burton and Stacey, 1984], the executive [Schoenfeld, 1986], which for most students has to
be brought to life, encouraged and strengthened in order
to get it operative It is the second bird described so
succinctly in the stanza from the Rig Veda [see for example O'Flaherty, 1982]:
Two birds clasp the self-same tree;
One eats of the sweet fruit, the other looks on
without eating
Even if this awareness-monitor is not present (since it
requires an attention split which is characteristic of
abstracting), the extending question is readily available to
the less sophisticated student since it is standard behaviour [for example Polya, 1970; Mason eta/, 1984; Mason,
1988] to try to place any result in a more general context.
But this sort of mathematical behaviour requires training
[Mason and Davis, 1987] precisely because it involves an
abstracting shift
Extending rephrased What sequences satisfy the
condition that In divides tn+ktn?
Comment. Note the momentary abstraction move which
this question signals The condition which involved a
struggle to express when first encountered in the sequence,
has become a property, akin to an axiom The search is
now on for other examples which satisfY this property
Possible ideas that people might try include other polynomials (how else might you generate a sequence?) Fibonacci and other similar sequences, geometric progressions, hyper-geometric progressions, etc . Not all of these
are satisfactory however
The sequence which was the goal of study in the beginning of the investigation comes to be seen as an object, a
particular instance of a class of similar objects, with similarity lying in the "divisibility criterion." Earlier I described abstraction as a shift of attention, from an expression of generality as the apex of a struggle to articulate, to
the expression as object 01 property. But simple reification is not quite adequate as a description The divisibility
property begins as foreground, as the object of attention,
the goal, and then, when expressed, recedes into the background and becomes a description, those specifications of
a property. The foreground becomes the sequences which
satisfy the property. Meaning is enriched not impoverished by the divorce of form from sensibility, because the
sensibility lies in the background, ready to be called upon
when required So in abstraction there is an element of
reification, and an element of change in stressing ignoring, of alteration of foreground and background, of
summarising rich experience in a distilled essence of form
During the event on which this article is based, someone observed that 3 not onlydivides21 (three terms on),
but the quotient is 7, which is the term following 3 The
person tried one other example in the head and then
announced a conjecture
Conjecture. Not only does tn divide subsequent
terms tn apart, but the quotient is also one of the
terms of the sequence. Check this for yomself, and
formulate a more precise conjecture
Comment There is a taste of abstraction in the shift hom
a pattern appearing in the numbers, to the pattern being
expressed and experienced as a conjecture Most students, upon detecting a pattern, dwell in the pattern and
its expression They may even invest self-respect in the
validity of their conjecture (manifested as a repeated
return to the conjecture in the face of other conjectures or
even of counter-examples), without experiencing their
pattern as a conjecture . The teacher has a role here, in
drawing attention to the status of the expression and
thereby inviting a helpful shift of attention, releasing
students from the confines of dwelling in the pattern
In the particular event on which this case study is based,
participants were attracted to this conjecture, so we
worked on expressing it and then looking for other
sequences that satisfy the same property Since my aim
was to focus attention on the shift from "sequence with a
property" to "property satisfied by what sequences" as a
generic example oft he shift abstraction in mathematics, I
was content to follow their inclination
As expected, we kept being drawn back to the expression In In+ 1 :: ln+ tn as people found that what they
thought they understood proved to be slippery The formula was written in blue on an OHP, with a blue box
around it, and frequently someone would interject with a
question such as "what are we assuming?" Someone else
would then say "the blue box formula" Many participants found the movement from verifying the formula on
the sequence 1, 3, 7, 13, , to looking for other sequences
satisfying the same condition an unstable shift
People proposed various polynomials, and we tried a
few simple ones such as ln = nand !n = n\ and concluded
that the property seemed only to apply to quadratics We
then tried to found out which quadratics, but the necessary algebra was too complicated for public calculation
on an OHP Later it was confirmed that the nontrivial
quadratic tn = an2 + bn + c satisfies
ln ln+J=fn+tn
if and only if a= c = 1 Here we have a clear example of the
mathematical force to characterise, seeking another condition or property which identifies those objects which
satisfy the given property In this case, the argument
requires considerable algebraic facility, including the
technique of equating coeffecients of nk in two different
polynomial expressions. This in itself requires an abstraction shift, from the equality of the values of two polynomials inn, to seeing the equality as a statement which must
hold for all n It requires a confidence with the coefficients
a, b and c as the objects under consideration, whereas,
until recently, it was then which was being investigated, a
shift of foreground and background mingled with allowing what were previously place-holders (a, b and c) to
become variables.
This is where the case study ends It had been a long
evening (this work being but part two of the session, and
some 80 people attending closely to what they were thinking and to what others were saying), and I felt that attention had been sufficiently sharply focused on the issue at
hand-the process of abstiacting
There is more to be learned from the exploration, however In preparing for the session, I had searched and
searched for some structural property characterising
sequences for which tn divides tn + ktn Starting from the
initial sequence a variety of different sequences with the
same property were found, but there were so many that it
was not at all obvious what to do next
One approach is to look at the structure of the set of
sequences, by seeing the sequences as objects, and looking
for operations on the sequences by which new objects can
be constructed (Note the etymological links between
structure and to construa) This is typically mathematical: drawing back (some would say abstracting) from the
examples, and considering the entire set of such sequences, together with operations on that set which preserve
the divisibility property. This led to a number of fascinating questions, but yielded little in the way of a characterisation, partly because of a lack of closure under term-wise
addition and multiplication Somehow more structure
was needed; an extra constraint had to be found
Returning to the original sequence, a further observation emerged, illustrating again the mathematical process
of abstraction Mathematical abstraction is intimately
connected with the mathematician's sense of structure,
which again seems to come down to that momentary shift
from an expression of generality, to the expression as
object to be manipulated. To illustrate this, consider the
following development of the case study
Observation. In the original sequence, terms two
) have their
places apart (e.g 1 and 7, 3 and 13,
difference divisible by two; terms three places apart
(e g. 1 and 13, 3 and 21) have their difference divisible by three. Does this always hold?
Comment. Again attention is drawn to finding out what
the thi.s actually is, by specialising, experimenting with
particular terms in the sequence, and trying to re-interpret
for yourself what general pattern was observed It is a
stronger statement even than the one reported earlier
involving d dividing t. implying that d divides t. +' The
rapidity of movement to a formula for tn, or even to a
5
general polynomial, is a measure of the confidence developed in the formula as an expression of generality and as
a manipulable object I conjecture that this is precisely
where sophisticated mathematician-teachers, unaware of
the momentary abstraction in themselves, miss the need
to attend to the abstracting movement in their students
Extending What sorts of sequences have this particular divisibility property?
Yet again there has been a shift, from the particular
sequence as context in which a property is being checked,
to considering all sequences satisfying that property I his
is again typical of the mathematician's move to characterise, which is so intimately bound up with abstraction The
very notion of characterising requires the abstraction
move to which I am trying to draw attention
Not all polynomials satisfy the property (for example,
triangular numbers) By looking for operations on such
sequences which preserve the property (such as finite
differences, deletion of initial terms, addition term by
term, multiplication term by term), conjectures and a
sense of structure begin to emerge. Asking whether operations such as finite difference and first-term-deletion can
be inverted leads to insight about the divisibility property
and what it implies
Note however the further acts of abstraction in switching from a few examples of sequences satisfying a property, to contemplating the set of sequences whose members satisfy a property, to verifying that a pair of
unknown sequences both satisfying a property can be
combined or opera ted upon to give yet another propertypreserving sequence To be able to treat a sequence sn as
an object with only one thing known about it, namely that
it satisfies a divisibility property, requires considerable
letting go of particularity and detail lt requires a true
drawing away from specificness. The experienced mathematician does this without thinking, and sometimes
leaves the novice gasping Mathematics teachers would
like their students also to make the delicate shift of
abstraction without thinking
The case study is at an end, but what pur pose has it
served? Has your attention been caught by the mathematical ideas, and if so, have you also observed the various
abstraction shifts which I claim are likely to be present?
This case study is no different from apparatus or hom any
mathematical example offered to students as an aid to
experiencing a generality. The very particularness of the
example or case study tends to attract attention to the
particularity The same is true of many computer programs The presence of a thing which is intended to be
generic, to act as a window through which the general is to
be experienced often does quite the opposite [Mason and
Pimm, 1984]. From my perspective, the case study illustrates or highlights certain features of mathematical
thinking to which I wished to draw attention. The same is
true of mathematical examples I o the teacher, they are
examples of some general idea, technique, principle or
theorem. I o students, they simply are. They are not
examples until they achieve examplehood
6
Most mathematics educators have a story about abstraction, and many have been attracted into mathematics education (in addition to their teaching) by the desire
to help students to cope with mathematical abstractions
Each new device, from Cuisenaire rods and geoboards to
Dienes blocks and dotty paper, to calculators and computer programs, is seen by succeeding generations as offering assistance. Thus for example, Kaput and PattisonGordon [1987] reported on extensive work with students
using computer programs designed to
ramp students from their concrete, situation-bound
thinking about the conceptual field of multiplicative structures to more abstract and flexible thinking
An important, perhaps defining, characteristic of these (computer) environments is their
systematic linking of concrete representations, beginning with iconic representations, to the more
abstract representations on which much of advanced
mathematics is built, e g. coordinate graphs and
algebraic functions This systematic linking of computer representations is the foundation of our strategy for building rich and flexible cognitive representations It is an attempt to build meaning and
understanding gradually in this conceptual field in a
way that will support long term competencecomputational and conceptual competence
One of the many features of computer programs, apart
from their dynamic and interactive qualities, is that they
present objects. Thus they can be used to support the
reification which is so fundamental to mathematical
thinking, though whether they then prove in some circumstances to focus attention so strongly on screen
objects that it becomes difficult to recognise the underlying form, is another question which needs a great deal of
research
Ever since HMI have written about school inspections
in the U.K, stress has been placed on the need for experience with apparatus [Mcintosh, 1981]. I hrough Dienes
[1964] it has become usual to speak of mathematics as
being embodied in apparatus, and this has come to imply
that mere contact with the apparatus will cause students
to experience mathematical ideas Hart [1989] has led an
attack on this myth, asking for direct evidence that apparatus actually helps. Unfortunately, embodiment is in the
eye of the beholder It is not the apparatus but how it is
perceived that matters Mathematicians, wishing to help
students, sometimes look for apparatus, for physical
objects (or physical situations, diagrams, images, or supposedly already familiar mathematical contexts) in which
they can see manifestations oftheir mathematical awareness, their abstraction They then offer their representation to students, perhaps forgetting that students may not
be seeing things from the same perspective. Students
stress and ignme according to their experience, and not
according to the experience of the expert. Thus it behoves
the expert to seek ways to draw attention to what is being
stressed, to help students to see through the particular
apparatus to a generality being exemplified, and then,
having gained confidence in articulating that generality,
to make the abstraction shift in which the generality
becomes object
In examining the issue of abstraction experienced by
undergraduates being taught linear algebra, Hare! [1987],
observes that embodiment is the reverse process to
abstraction (and hence embodiment-abstraction is analogous to specialising-generalising) Looking at textbooks, he found that all but one author offered examples
of vector spaces (at various levels of generality I abstraction), and then launched into abstract definition with
little or no reference to the examples except as later
"checks "It is as if authors fail to recognise the fine detail
of what students actually need. Only Pedoe [1969] defined
the same technical term three times, in connection with
three examples at three different levels of generality and
abstraction Once abstraction is recognised as a shift of
attention, it is possible to be of direct assistance to students, by constructing activities which call upon their
powers of generalising to express in their own terms what
is the same about a number of, for them, situations, for
the author, examples, and then explicitly draw attention
to the movement in which the expressions become
manipulable entities. Advice can be offered about how to
prepare for this move, and mathematical challenges can
be offered which attract attention away from the generalising, and so help students to subordinate the generalisation aspect of the expressions through integrating it into
their thinking [Gattegno, 1987] and so participate in or
experience the abstracting
The notion of embodiment is encircled by the teacher's
dilemma of not knowing what it is that the students are
attending to. Frequently teachers are not aware of what
they are attending to either, in the sense that they are not
aware that they are stressing and ignoring, nor that their
students may not be stressing the same features. Being
a ware of the fact of stressing and ignoring, in the midst of
a teaching moment, enables a teacher to focus attention
on, to stress explicitly, the act of stressing
Traditionally, mathematics has been presented to students in a straightforward, rehearsal mode, in which the
expert rehearses publicly the honed presentation of theorems, definitions, proofs and examples Those who begin
to make the move to abstraction spontaneously go on to
study mathematics fmther, while the others remain mystified, though sometimes intrigued. Exposition is cmrently
blamed for most of the failures of mathematics teaching,
but it is not the mode or style which is responsible
Rather, it is the location of the teacher's attention, and the
extent of the teachers' mathematical being which determines what happens within any mode of teacher-studentcontent interaction [Mason, 1979] Despite cunent emphasis on exploration and investigation, many students
may still not expedence the shift of abstraction unless
they receive explicit assistance. By being explicit, by
focussing attention on the brief but important moments
of abstraction movement, more students could be helped
to appreciate the power and pleasure of mathematics, and
to get more from their investigating. One of the roles of a
teacher is to help students to draw back from detail, to
shift into a more reflective contemplation of what they are
doing, to keep track of the large picture while students
pursue details This is what Vygotsky signalled with his
zone of proximal development and the role of the teacher
in providing an extended awareness [Bruner, 1986]
Dubinsky and l ewin [1986] found difficulty in prompting students to abstract. Using the language of Piaget,
they claim that
the most powerful, and cognitively, the most interesting form of equilibration is that in which particular cognitive structures re-equilibrate to a disturbance by undergoing a greater or lesser degree of
re-construction, a process known as "reflective
abstraction " [p 61]
After discussing two mathematical topics which pose
major hurdles to undergraduates, they conclude that one
of the most serious obstacles to overcome is the propensity of students
to persist in applying an incorrect understanding of
a concept even after a teacher has pointed out the
error and tried to re-teach the concept
students
will accept the criticism and instruction overtly, but
revert to inco11ect usage as soon as possible, somewhat amused by the teacher's apparent pedantry
[p 88]
Dubinsky and l ewin suggest that either the student has
no overall sense of the target concept, or has an integrated
but incorrect concept, and that disequilibration can only
take place when the student realises that thinking can be
subject to disconfirmation and alteration. Thus, Dubinsky and Lewin seem to be invoking a metacognitive shift
in awareness in order to bring about a cognitive shift in
perspective Such metacognitive shifts are not all that
difficult to bring about, being best supported by undertaking to establish a mathematical, conjecturing atmosphere in classes, where what is said is assumed to be said in
order that it may be subsequently modified[Polya, 1970;
Mason et a/, 1984; Mason, 1988].
But there are other explanations for the robustness of
current ways of thinking, and other ways supporting a
cognitive shift of attention . It is not just student wilfulness that makes teaching the frustrating task implied in
the previous quotation. 1he didactic tension, or the
didactic transposition as Brousseau [1984] calls it, is
always present:
The more explicit I am about the behaviour I wish
my students to display, the more likely it is that they
will display that behaviour without recourse to the
understanding which the behaviour is meant to
indicate; that is, the more they will take the form for
the substance
Students are interested, as are we all, in minimising the
energy they need to invest in order to get through events
Thus they look for cues, in what the teacher says and does,
for what they should do The result is often a superficial
production of behaviour, without roots adequate to enable that behaviour to be modified and adapted in slightly
different circumstances (as when the questions are posed
7
slightly differently on a test) Behaviour is trainable, but
training alone is not enough to secure flexibility and
competence [Mason, 1989)
A temporary shift can take place through students
producing the behaviour being demanded, but a robust
shift of attention, or rather a flexibility to be able to
attend in more than one way, is what I believe Dubinsky
and Lewin mean by the term reflective abstraction By
working on the locus and structure of their own attention,
and through that, on students' attention, teachers can
contribute to a more rounded and balanced experience
for students than a narrow focus on training behaviour
There are strategies for harnessing emotion and educating
awareness as well as training behaviour [see fm example
Griffin, 1989) When computer programs are constructed
to illustrate or manifest mathematical ideas, we have an
example of stressing made manifest But the program by
itself is not sufficient for most students They also need
help in construing what they see and do, weaving it into a
story in their own words [Mason, 1985]
The helix depicted earlier was introduced precisely in
order to assist teachers to notice opportunities to help
students in shifts of attention such as abstracting. It is
intended to provide an image, linking the labels manipulating, getting-a-sense-of, articulating with personal experience, and so awakening a teacher to the value in noting
such transitions explicitly and allowing students time to
convert hard-won expressions of generality into confidently manipulable objects There is no doubt that such
frameworks help [Mason and Davis, 1987), and indeed
may in the end be the single most effective tool for helping
teachers to develop their teaching and their sense of what
it means to be mathematical, and thereby helping students to experience the delicate shifts of attention which
mathematicians call abstraction
Refel'ences
Bruner. .Jerome[l986] Auua/mmds pomblf worlds Harvard University Press, Cambridge, Mass
Brousseau, Guy [1984] The Crucial Role of the Didactical Contract in
the Analysis and Construction of Situations in Teaching and learning. Proc of the Thwr} of Maths Eduwtwn Group ICME 5. lost
ftir Didaktik der Math Occasional Paper 54 Bielefeld
Dienes, Zoltan [1964] 1he power of mat hematin Hutchinson, New
York
Dubinksy Ed & Lewin, Philip [1986] Reflective Abstraction and
Mathematics Education: the Genetic Decomposition of Induction
and Compactness Journal of Mathematiwl Behavior 5: 55-92
Eco, Umberto [1988] 1he aer,theti!s of S't Thomas Aquinao, (trans H
Bredin) Radius, London
Floyd . Ann; a a/ [1982] EM23 Developing mathematiwl thinking
Open University undergraduate course Open University, Milton
Keynes
8
Gattegno, Caleb [1987] The sden(e of cduwtion Pan I theoretical
wmidaat1ons. Educational Solutions, New York
Griffin Pete[l989] PM643A Prcpmingto Temh Angle Open University, Milton Keynes
Hare!. Guershon [1987] Variations in Linear Algebra Content Presentations For the Learning of MathematiC\ 7 (3); 29-32
Hart, Kath [1989] There is little Connection In P Ernest (ed) Mathematics temhing the state of the art Falmer Press, Lewes . pp
138-142
Honsberger, Ross [1970] !ngenuin in mathematus Mathematical
Association of America, Washington
Kaput, James & Pattison-Gordon. Laurie [1987] A conuete-to-abstw! t
sojtYvare ramp environments for lwrning mulliplhation. div1o,ion
and intensive quantity Word Problems Project 1 echnical Report
Harvard University
Mason. John [1979] Which Medium Which Message? Visual Education Feb p 29-33
Mason John [1985] What do you do when You Turn Off the Machine!
In M. Pluvinage(ed) 1heinfluwc;:_ ofwmputen andmformatics on
mathematio and its tea(hing lost de Recherche sur l · Enseignement des Matht:matiques. Strasbourg
Mason John [1989] What Exactly can a Teacher Do? (submitted)
Mason. John & Davis, Joy [1987] The Use of Explicitly Introduced
Vocabulary in Helping Students to Learn and Teachers to teach
Mathematics Proceeding5 of PM E-X I (ed. J Bergeron, N Herscovics & C. Kieran) val. 3: 275-281. Montreal
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Pro(etdings of PM E~ )(// (ed A Barbas) vol 2: 487-494 Vezprem
Hungary
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Mathemali(S, 1(2): 8-12
Mason, John [1984] Towards One Possible Discipline of Mathematics
Education. Theor} and Pra(f/(e of Mathematic~ Eduwtiun, Occasional Paper 54. lost ftit Didaktik der Math der U Bielefeld: 42-45
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Mason, John [1988] Learning and doing mathemano Macmillan
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Mcintosh, Alistair [1981] When Will They Ever Learn? In Floyd (ed)
Devtloping mathematical thmking Addison Wesley: 6-11
0 Flaherty, W [1982] The Rig Veda. Book I Hymn 164 Verse 20
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Wiley, New York
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Indiana University Press, Bloomington
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Press
St Augustine[389] De Magi1tro(trans G leckie) Appleton-CenturyCroft
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